modelparameters.sympy.polys.domains package

Submodules

modelparameters.sympy.polys.domains.algebraicfield module

Implementation of AlgebraicField class.

class modelparameters.sympy.polys.domains.algebraicfield.AlgebraicField(dom, *ext)

Bases: Field, CharacteristicZero, SimpleDomain

A class for representing algebraic number fields.

algebraic_field(*extension)

Returns an algebraic field, i.e. mathbb{Q}(alpha, ldots).

denom(a)

Returns denominator of a.

dtype

alias of ANP

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s expression to dtype.

get_ring()

Returns a ring associated with self.

has_assoc_Field = True

has_assoc_Ring = False

is_Algebraic = True

is_AlgebraicField = True

is_Numerical = True

is_negative(a)

Returns True if a is negative.

is_nonnegative(a)

Returns True if a is non-negative.

is_nonpositive(a)

Returns True if a is non-positive.

is_positive(a)

Returns True if a is positive.

new(element)

numer(a)

Returns numerator of a.

to_sympy(a)

Convert a to a SymPy object.

modelparameters.sympy.polys.domains.characteristiczero module

Implementaton of CharacteristicZero class.

class modelparameters.sympy.polys.domains.characteristiczero.CharacteristicZero

Bases: Domain

Domain that has infinite number of elements.

characteristic()

Return the characteristic of this domain.

has_CharacteristicZero = True

modelparameters.sympy.polys.domains.complexfield module

Implementation of ComplexField class.

class modelparameters.sympy.polys.domains.complexfield.ComplexField(prec=53, dps=None, tol=None)

Bases: Field, CharacteristicZero, SimpleDomain

Complex numbers up to the given precision.

almosteq(a, b, tolerance=None)

Check if a and b are almost equal.

property dps

from_ComplexField(element, base)

Convert a complex element to dtype.

from_QQ_gmpy(element, base)

Convert a GMPY mpq object to dtype.

from_QQ_python(element, base)

Convert a Python Fraction object to dtype.

from_RealField(element, base)

Convert a real element object to dtype.

from_ZZ_gmpy(element, base)

Convert a GMPY mpz object to dtype.

from_ZZ_python(element, base)

Convert a Python int object to dtype.

from_sympy(expr)

Convert SymPy’s number to dtype.

gcd(a, b)

Returns GCD of a and b.

get_exact()

Returns an exact domain associated with self.

get_ring()

Returns a ring associated with self.

has_assoc_Field = True

has_assoc_Ring = False

property has_default_precision

is_CC = True

is_ComplexField = True

is_Exact = False

is_Numerical = True

lcm(a, b)

Returns LCM of a and b.

property precision

rep = 'CC'

to_sympy(element)

Convert element to SymPy number.

property tolerance

modelparameters.sympy.polys.domains.compositedomain module

Implementation of CompositeDomain class.

class modelparameters.sympy.polys.domains.compositedomain.CompositeDomain

Bases: Domain

Base class for composite domains, e.g. ZZ[x], ZZ(X).

domain = None

gens = None

inject(*symbols)

Inject generators into this domain.

is_Composite = True

ngens = None

symbols = None

modelparameters.sympy.polys.domains.domain module

Implementation of Domain class.

class modelparameters.sympy.polys.domains.domain.Domain

Bases: object

Represents an abstract domain.

abs(a)

Absolute value of a, implies __abs__.

add(a, b)

Sum of a and b, implies __add__.

algebraic_field(*extension)

Returns an algebraic field, i.e. K(alpha, ldots).

alias = None

almosteq(a, b, tolerance=None)

Check if a and b are almost equal.

characteristic()

Return the characteristic of this domain.

cofactors(a, b)

Returns GCD and cofactors of a and b.

convert(element, base=None)

Convert element to self.dtype.

convert_from(element, base)

Convert element to self.dtype given the base domain.

denom(a)

Returns denominator of a.

div(a, b)

Division of a and b, implies something.

dtype = None

evalf(a, prec=None, **options)

Returns numerical approximation of a.

exquo(a, b)

Exact quotient of a and b, implies something.

frac_field(*symbols, **kwargs)

Returns a fraction field, i.e. K(X).

from_AlgebraicField(a, K0)

Convert an algebraic number to dtype.

from_ComplexField(a, K0)

Convert a complex element to dtype.

from_ExpressionDomain(a, K0)

Convert a EX object to dtype.

from_FF_gmpy(a, K0)

Convert ModularInteger(mpz) to dtype.

from_FF_python(a, K0)

Convert ModularInteger(int) to dtype.

from_FractionField(a, K0)

Convert a rational function to dtype.

from_GeneralizedPolynomialRing(a, K0)

from_GlobalPolynomialRing(a, K0)

Convert a polynomial to dtype.

from_PolynomialRing(a, K0)

Convert a polynomial to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a real element object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert a SymPy object to dtype.

gcd(a, b)

Returns GCD of a and b.

gcdex(a, b)

Extended GCD of a and b.

get_exact()

Returns an exact domain associated with self.

get_field()

Returns a field associated with self.

get_ring()

Returns a ring associated with self.

half_gcdex(a, b)

Half extended GCD of a and b.

has_CharacteristicZero = False

property has_Field

property has_Ring

has_assoc_Field = False

has_assoc_Ring = False

imag(a)

inject(*symbols)

Inject generators into this domain.

invert(a, b)

Returns inversion of a mod b, implies something.

is_Algebraic = False

is_AlgebraicField = False

is_CC = False

is_ComplexField = False

is_Composite = False

is_EX = False

is_Exact = True

is_FF = False

is_Field = False

is_FiniteField = False

is_Frac = False

is_FractionField = False

is_IntegerRing = False

is_Numerical = False

is_PID = False

is_Poly = False

is_PolynomialRing = False

is_QQ = False

is_RR = False

is_RationalField = False

is_RealField = False

is_Ring = False

is_Simple = False

is_SymbolicDomain = False

is_ZZ = False

is_negative(a)

Returns True if a is negative.

is_nonnegative(a)

Returns True if a is non-negative.

is_nonpositive(a)

Returns True if a is non-positive.

is_one(a)

Returns True if a is one.

is_positive(a)

Returns True if a is positive.

is_zero(a)

Returns True if a is zero.

lcm(a, b)

Returns LCM of a and b.

log(a, b)

Returns b-base logarithm of a.

map(seq)

Rersively apply self to all elements of seq.

mul(a, b)

Product of a and b, implies __mul__.

n(a, prec=None, **options)

Returns numerical approximation of a.

neg(a)

Returns a negated, implies __neg__.

new(*args)

normal(*args)

numer(a)

Returns numerator of a.

of_type(element)

Check if a is of type dtype.

old_frac_field(*symbols, **kwargs)

Returns a fraction field, i.e. K(X).

old_poly_ring(*symbols, **kwargs)

Returns a polynomial ring, i.e. K[X].

one = None

poly_ring(*symbols, **kwargs)

Returns a polynomial ring, i.e. K[X].

pos(a)

Returns a positive, implies __pos__.

pow(a, b)

Raise a to power b, implies __pow__.

quo(a, b)

Quotient of a and b, implies something.

real(a)

rem(a, b)

Remainder of a and b, implies __mod__.

rep = None

revert(a)

Returns a**(-1) if possible.

sqrt(a)

Returns square root of a.

sub(a, b)

Difference of a and b, implies __sub__.

to_sympy(a)

Convert a to a SymPy object.

property tp

unify(K1, symbols=None)

Construct a minimal domain that contains elements of K0 and K1.

Known domains (from smallest to largest):

• GF(p)

• ZZ

• QQ

• RR(prec, tol)

• CC(prec, tol)

• ALG(a, b, c)

• K[x, y, z]

• K(x, y, z)

• EX

unify_with_symbols(K1, symbols)

zero = None

modelparameters.sympy.polys.domains.domainelement module

Trait for implementing domain elements.

class modelparameters.sympy.polys.domains.domainelement.DomainElement

Bases: object

Represents an element of a domain.

Mix in this trait into a class which instances should be recognized as elements of a domain. Method parent() gives that domain.

parent()

modelparameters.sympy.polys.domains.expressiondomain module

Implementation of ExpressionDomain class.

class modelparameters.sympy.polys.domains.expressiondomain.ExpressionDomain

Bases: Field, CharacteristicZero, SimpleDomain

A class for arbitrary expressions.

class Expression(ex)

Bases: PicklableWithSlots

An arbitrary expression.

as_expr()

denom()

ex

gcd(g)

lcm(g)

numer()

simplify(ex)

denom(a)

Returns denominator of a.

dtype

alias of Expression

from_ExpressionDomain(a, K0)

Convert a EX object to dtype.

from_FractionField(a, K0)

Convert a DMF object to dtype.

from_PolynomialRing(a, K0)

Convert a DMP object to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s expression to dtype.

gcd(a, b)

Returns GCD of a and b.

This definition of GCD over fields allows to clear denominators in primitive().

>>> from ..domains import QQ
>>> from ... import S, gcd, primitive
>>> from ...abc import x

>>> QQ.gcd(QQ(2, 3), QQ(4, 9))
2/9
>>> gcd(S(2)/3, S(4)/9)
2/9
>>> primitive(2*x/3 + S(4)/9)
(2/9, 3*x + 2)

get_field()

Returns a field associated with self.

get_ring()

Returns a ring associated with self.

has_assoc_Field = True

has_assoc_Ring = False

is_EX = True

is_SymbolicDomain = True

is_negative(a)

Returns True if a is negative.

is_nonnegative(a)

Returns True if a is non-negative.

is_nonpositive(a)

Returns True if a is non-positive.

is_positive(a)

Returns True if a is positive.

lcm(a, b)

Returns LCM of a and b.

>>> from ..domains import QQ
>>> from ... import S, lcm

>>> QQ.lcm(QQ(2, 3), QQ(4, 9))
4/3
>>> lcm(S(2)/3, S(4)/9)
4/3

numer(a)

Returns numerator of a.

one = EX(1)

rep = 'EX'

to_sympy(a)

Convert a to a SymPy object.

zero = EX(0)

modelparameters.sympy.polys.domains.field module

Implementation of Field class.

class modelparameters.sympy.polys.domains.field.Field

Bases: Ring

Represents a field domain.

div(a, b)

Division of a and b, implies __div__.

exquo(a, b)

Exact quotient of a and b, implies __div__.

gcd(a, b)

Returns GCD of a and b.

This definition of GCD over fields allows to clear denominators in primitive().

>>> from ..domains import QQ
>>> from ... import S, gcd, primitive
>>> from ...abc import x

>>> QQ.gcd(QQ(2, 3), QQ(4, 9))
2/9
>>> gcd(S(2)/3, S(4)/9)
2/9
>>> primitive(2*x/3 + S(4)/9)
(2/9, 3*x + 2)

get_field()

Returns a field associated with self.

get_ring()

Returns a ring associated with self.

is_Field = True

is_PID = True

lcm(a, b)

Returns LCM of a and b.

>>> from ..domains import QQ
>>> from ... import S, lcm

>>> QQ.lcm(QQ(2, 3), QQ(4, 9))
4/3
>>> lcm(S(2)/3, S(4)/9)
4/3

quo(a, b)

Quotient of a and b, implies __div__.

rem(a, b)

Remainder of a and b, implies nothing.

revert(a)

Returns a**(-1) if possible.

modelparameters.sympy.polys.domains.finitefield module

Implementation of FiniteField class.

class modelparameters.sympy.polys.domains.finitefield.FiniteField(mod, dom=None, symmetric=True)

Bases: Field, SimpleDomain

General class for finite fields.

characteristic()

Return the characteristic of this domain.

dom = None

from_FF_gmpy(a, K0=None)

Convert ModularInteger(mpz) to dtype.

from_FF_python(a, K0=None)

Convert ModularInteger(int) to dtype.

from_QQ_gmpy(a, K0=None)

Convert GMPY’s mpq to dtype.

from_QQ_python(a, K0=None)

Convert Python’s Fraction to dtype.

from_RealField(a, K0)

Convert mpmath’s mpf to dtype.

from_ZZ_gmpy(a, K0=None)

Convert GMPY’s mpz to dtype.

from_ZZ_python(a, K0=None)

Convert Python’s int to dtype.

from_sympy(a)

Convert SymPy’s Integer to SymPy’s Integer.

get_field()

Returns a field associated with self.

has_assoc_Field = True

has_assoc_Ring = False

is_FF = True

is_FiniteField = True

is_Numerical = True

mod = None

rep = 'FF'

to_sympy(a)

Convert a to a SymPy object.

modelparameters.sympy.polys.domains.fractionfield module

Implementation of FractionField class.

class modelparameters.sympy.polys.domains.fractionfield.FractionField(domain_or_field, symbols=None, order=None)

Bases: Field, CompositeDomain

A class for representing multivariate rational function fields.

denom(a)

Returns denominator of a.

factorial(a)

Returns factorial of a.

from_AlgebraicField(a, K0)

Convert an algebraic number to dtype.

from_FractionField(a, K0)

Convert a rational function to dtype.

from_PolynomialRing(a, K0)

Convert a polynomial to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s expression to dtype.

get_ring()

Returns a field associated with self.

has_assoc_Field = True

has_assoc_Ring = True

is_Frac = True

is_FractionField = True

is_negative(a)

Returns True if LC(a) is negative.

is_nonnegative(a)

Returns True if LC(a) is non-negative.

is_nonpositive(a)

Returns True if LC(a) is non-positive.

is_positive(a)

Returns True if LC(a) is positive.

new(element)

numer(a)

Returns numerator of a.

property one

property order

to_sympy(a)

Convert a to a SymPy object.

property zero

modelparameters.sympy.polys.domains.gmpyfinitefield module

Implementation of GMPYFiniteField class.

class modelparameters.sympy.polys.domains.gmpyfinitefield.GMPYFiniteField(mod, symmetric=True)

Bases: FiniteField

Finite field based on GMPY integers.

alias = 'FF_gmpy'

modelparameters.sympy.polys.domains.gmpyintegerring module

Implementaton of GMPYIntegerRing class.

class modelparameters.sympy.polys.domains.gmpyintegerring.GMPYIntegerRing

Bases: IntegerRing

Integer ring based on GMPY’s mpz type.

alias = 'ZZ_gmpy'

dtype

alias of GMPYInteger

factorial(a)

Compute factorial of a.

from_FF_gmpy(a, K0)

Convert ModularInteger(mpz) to GMPY’s mpz.

from_FF_python(a, K0)

Convert ModularInteger(int) to GMPY’s mpz.

from_QQ_gmpy(a, K0)

Convert GMPY mpq to GMPY’s mpz.

from_QQ_python(a, K0)

Convert Python’s Fraction to GMPY’s mpz.

from_RealField(a, K0)

Convert mpmath’s mpf to GMPY’s mpz.

from_ZZ_gmpy(a, K0)

Convert GMPY’s mpz to GMPY’s mpz.

from_ZZ_python(a, K0)

Convert Python’s int to GMPY’s mpz.

from_sympy(a)

Convert SymPy’s Integer to dtype.

gcd(a, b)

Compute GCD of a and b.

gcdex(a, b)

Compute extended GCD of a and b.

lcm(a, b)

Compute LCM of a and b.

one = <modelparameters.sympy.polys.domains.groundtypes.GMPYInteger object>

sqrt(a)

Compute square root of a.

to_sympy(a)

Convert a to a SymPy object.

tp

alias of GMPYInteger

zero = <modelparameters.sympy.polys.domains.groundtypes.GMPYInteger object>

modelparameters.sympy.polys.domains.gmpyrationalfield module

Implementaton of GMPYRationalField class.

class modelparameters.sympy.polys.domains.gmpyrationalfield.GMPYRationalField

Bases: RationalField

Rational field based on GMPY mpq class.

alias = 'QQ_gmpy'

denom(a)

Returns denominator of a.

div(a, b)

Division of a and b, implies __div__.

dtype

alias of GMPYRational

exquo(a, b)

Exact quotient of a and b, implies __div__.

factorial(a)

Returns factorial of a.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s Integer to dtype.

get_ring()

Returns ring associated with self.

numer(a)

Returns numerator of a.

one = <modelparameters.sympy.polys.domains.groundtypes.GMPYRational object>

quo(a, b)

Quotient of a and b, implies __div__.

rem(a, b)

Remainder of a and b, implies nothing.

to_sympy(a)

Convert a to a SymPy object.

tp

alias of GMPYRational

zero = <modelparameters.sympy.polys.domains.groundtypes.GMPYRational object>

modelparameters.sympy.polys.domains.groundtypes module

Ground types for various mathematical domains in SymPy.

modelparameters.sympy.polys.domains.integerring module

Implementation of IntegerRing class.

class modelparameters.sympy.polys.domains.integerring.IntegerRing

Bases: Ring, CharacteristicZero, SimpleDomain

General class for integer rings.

algebraic_field(*extension)

Returns an algebraic field, i.e. mathbb{Q}(alpha, ldots).

from_AlgebraicField(a, K0)

Convert a ANP object to dtype.

get_field()

Returns a field associated with self.

has_assoc_Field = True

has_assoc_Ring = True

is_IntegerRing = True

is_Numerical = True

is_PID = True

is_ZZ = True

log(a, b)

Returns b-base logarithm of a.

rep = 'ZZ'

modelparameters.sympy.polys.domains.modularinteger module

Implementation of ModularInteger class.

class modelparameters.sympy.polys.domains.modularinteger.ModularInteger(val)

Bases: PicklableWithSlots, DomainElement

A class representing a modular integer.

dom = None

invert()

mod = None

parent()

sym = None

to_int()

val

modelparameters.sympy.polys.domains.mpelements module

Real and complex elements.

class modelparameters.sympy.polys.domains.mpelements.ComplexElement(real=0, imag=0)

Bases: _mpc, DomainElement

An element of a complex domain.

context = <modelparameters.sympy.polys.domains.mpelements.MPContext object>

parent()

class modelparameters.sympy.polys.domains.mpelements.MPContext(prec=53, dps=None, tol=None)

Bases: PythonMPContext

almosteq(s, t, rel_eps=None, abs_eps=None)

make_tol()

to_rational(s, limit=True)

class modelparameters.sympy.polys.domains.mpelements.RealElement(val=(0, 0, 0, 0), **kwargs)

Bases: _mpf, DomainElement

An element of a real domain.

context = <modelparameters.sympy.polys.domains.mpelements.MPContext object>

parent()

modelparameters.sympy.polys.domains.old_fractionfield module

Implementation of FractionField class.

class modelparameters.sympy.polys.domains.old_fractionfield.FractionField(dom, *gens)

Bases: Field, CharacteristicZero, CompositeDomain

A class for representing rational function fields.

denom(a)

Returns denominator of a.

dtype

alias of DMF

factorial(a)

Returns factorial of a.

frac_field(*gens)

Returns a fraction field, i.e. K(X).

from_FractionField(a, K0)

Convert a fraction field element to another fraction field.

Examples

>>> from ..polyclasses import DMF
>>> from ..domains import ZZ, QQ
>>> from ...abc import x

>>> f = DMF(([ZZ(1), ZZ(2)], [ZZ(1), ZZ(1)]), ZZ)

>>> QQx = QQ.old_frac_field(x)
>>> ZZx = ZZ.old_frac_field(x)

>>> QQx.from_FractionField(f, ZZx)
(x + 2)/(x + 1)

from_GlobalPolynomialRing(a, K0)

Convert a DMF object to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s expression to dtype.

get_ring()

Returns a ring associated with self.

has_assoc_Field = True

has_assoc_Ring = True

is_Frac = True

is_FractionField = True

is_negative(a)

Returns True if a is negative.

is_nonnegative(a)

Returns True if a is non-negative.

is_nonpositive(a)

Returns True if a is non-positive.

is_positive(a)

Returns True if a is positive.

new(element)

numer(a)

Returns numerator of a.

poly_ring(*gens)

Returns a polynomial ring, i.e. K[X].

to_sympy(a)

Convert a to a SymPy object.

modelparameters.sympy.polys.domains.old_polynomialring module

Implementation of PolynomialRing class.

class modelparameters.sympy.polys.domains.old_polynomialring.GlobalPolynomialRing(dom, *gens, **opts)

Bases: PolynomialRingBase

A true polynomial ring, with objects DMP.

dtype

alias of DMP

from_FractionField(a, K0)

Convert a DMF object to DMP.

Examples

>>> from ..polyclasses import DMP, DMF
>>> from ..domains import ZZ
>>> from ...abc import x

>>> f = DMF(([ZZ(1), ZZ(1)], [ZZ(1)]), ZZ)
>>> K = ZZ.old_frac_field(x)

>>> F = ZZ.old_poly_ring(x).from_FractionField(f, K)

>>> F == DMP([ZZ(1), ZZ(1)], ZZ)
True
>>> type(F)
<class 'sympy.polys.polyclasses.DMP'>

from_sympy(a)

Convert SymPy’s expression to dtype.

is_Poly = True

is_PolynomialRing = True

is_negative(a)

Returns True if LC(a) is negative.

is_nonnegative(a)

Returns True if LC(a) is non-negative.

is_nonpositive(a)

Returns True if LC(a) is non-positive.

is_positive(a)

Returns True if LC(a) is positive.

to_sympy(a)

Convert a to a SymPy object.

modelparameters.sympy.polys.domains.old_polynomialring.PolynomialRing(dom, *gens, **opts)

Create a generalized multivariate polynomial ring.

A generalized polynomial ring is defined by a ground field K, a set of generators (typically x_1, ldots, x_n) and a monomial order <. The monomial order can be global, local or mixed. In any case it induces a total ordering on the monomials, and there exists for every (non-zero) polynomial f in K[x_1, ldots, x_n] a well-defined “leading monomial” LM(f) = LM(f, >). One can then define a multiplicative subset S = S_> = {f in K[x_1, ldots, x_n] | LM(f) = 1}. The generalized polynomial ring corresponding to the monomial order is R = S^{-1}K[x_1, ldots, x_n].

If > is a so-called global order, that is 1 is the smallest monomial, then we just have S = K and R = K[x_1, ldots, x_n].

Examples

A few examples may make this clearer.

>>> from ...abc import x, y
>>> from ... import QQ

Our first ring uses global lexicographic order.

>>> R1 = QQ.old_poly_ring(x, y, order=(("lex", x, y),))

The second ring uses local lexicographic order. Note that when using a single (non-product) order, you can just specify the name and omit the variables:

>>> R2 = QQ.old_poly_ring(x, y, order="ilex")

The third and fourth rings use a mixed orders:

>>> o1 = (("ilex", x), ("lex", y))
>>> o2 = (("lex", x), ("ilex", y))
>>> R3 = QQ.old_poly_ring(x, y, order=o1)
>>> R4 = QQ.old_poly_ring(x, y, order=o2)

We will investigate what elements of K(x, y) are contained in the various rings.

>>> L = [x, 1/x, y/(1 + x), 1/(1 + y), 1/(1 + x*y)]
>>> test = lambda R: [f in R for f in L]

The first ring is just K[x, y]:

>>> test(R1)
[True, False, False, False, False]

The second ring is R1 localised at the maximal ideal (x, y):

>>> test(R2)
[True, False, True, True, True]

The third ring is R1 localised at the prime ideal (x):

>>> test(R3)
[True, False, True, False, True]

Finally the fourth ring is R1 localised at S = K[x, y] setminus yK[y]:

>>> test(R4)
[True, False, False, True, False]

class modelparameters.sympy.polys.domains.old_polynomialring.PolynomialRingBase(dom, *gens, **opts)

Bases: Ring, CharacteristicZero, CompositeDomain

Base class for generalized polynomial rings.

This base class should be used for uniform access to generalized polynomial rings. Subclasses only supply information about the element storage etc.

Do not instantiate.

default_order = 'grevlex'

factorial(a)

Returns factorial of a.

frac_field(*gens)

Returns a fraction field, i.e. K(X).

free_module(rank)

Generate a free module of rank rank over self.

>>> from ...abc import x
>>> from ... import QQ
>>> QQ.old_poly_ring(x).free_module(2)
QQ[x]**2

from_AlgebraicField(a, K0)

Convert a ANP object to dtype.

from_GlobalPolynomialRing(a, K0)

Convert a DMP object to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

gcd(a, b)

Returns GCD of a and b.

gcdex(a, b)

Extended GCD of a and b.

get_field()

Returns a field associated with self.

has_assoc_Field = True

has_assoc_Ring = True

lcm(a, b)

Returns LCM of a and b.

new(element)

poly_ring(*gens)

Returns a polynomial ring, i.e. K[X].

revert(a)

Returns a**(-1) if possible.

modelparameters.sympy.polys.domains.polynomialring module

Implementation of PolynomialRing class.

class modelparameters.sympy.polys.domains.polynomialring.PolynomialRing(domain_or_ring, symbols=None, order=None)

Bases: Ring, CompositeDomain

A class for representing multivariate polynomial rings.

factorial(a)

Returns factorial of a.

from_AlgebraicField(a, K0)

Convert an algebraic number to dtype.

from_FractionField(a, K0)

Convert a rational function to dtype.

from_PolynomialRing(a, K0)

Convert a polynomial to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s expression to dtype.

gcd(a, b)

Returns GCD of a and b.

gcdex(a, b)

Extended GCD of a and b.

get_field()

Returns a field associated with self.

has_assoc_Field = True

has_assoc_Ring = True

is_Poly = True

is_PolynomialRing = True

is_negative(a)

Returns True if LC(a) is negative.

is_nonnegative(a)

Returns True if LC(a) is non-negative.

is_nonpositive(a)

Returns True if LC(a) is non-positive.

is_positive(a)

Returns True if LC(a) is positive.

lcm(a, b)

Returns LCM of a and b.

new(element)

property one

property order

to_sympy(a)

Convert a to a SymPy object.

property zero

modelparameters.sympy.polys.domains.pythonfinitefield module

Implementation of PythonFiniteField class.

class modelparameters.sympy.polys.domains.pythonfinitefield.PythonFiniteField(mod, symmetric=True)

Bases: FiniteField

Finite field based on Python’s integers.

alias = 'FF_python'

modelparameters.sympy.polys.domains.pythonintegerring module

Implementaton of PythonIntegerRing class.

class modelparameters.sympy.polys.domains.pythonintegerring.PythonIntegerRing

Bases: IntegerRing

Integer ring based on Python’s int type.

alias = 'ZZ_python'

dtype

alias of int

factorial(a)

Compute factorial of a.

from_FF_gmpy(a, K0)

Convert ModularInteger(mpz) to Python’s int.

from_FF_python(a, K0)

Convert ModularInteger(int) to Python’s int.

from_QQ_gmpy(a, K0)

Convert GMPY’s mpq to Python’s int.

from_QQ_python(a, K0)

Convert Python’s Fraction to Python’s int.

from_RealField(a, K0)

Convert mpmath’s mpf to Python’s int.

from_ZZ_gmpy(a, K0)

Convert GMPY’s mpz to Python’s int.

from_ZZ_python(a, K0)

Convert Python’s int to Python’s int.

from_sympy(a)

Convert SymPy’s Integer to dtype.

gcd(a, b)

Compute GCD of a and b.

gcdex(a, b)

Compute extended GCD of a and b.

lcm(a, b)

Compute LCM of a and b.

one = 1

sqrt(a)

Compute square root of a.

to_sympy(a)

Convert a to a SymPy object.

zero = 0

modelparameters.sympy.polys.domains.pythonrational module

Rational number type based on Python integers.

class modelparameters.sympy.polys.domains.pythonrational.PythonRational(p, q=1, _gcd=True)

Bases: DefaultPrinting, PicklableWithSlots, DomainElement

Rational number type based on Python integers.

This was supposed to be needed for compatibility with older Python versions which don’t support Fraction. However, Fraction is very slow so we don’t use it anyway.

Examples

>>> from ..domains import PythonRational

>>> PythonRational(1)
1
>>> PythonRational(2, 3)
2/3
>>> PythonRational(14, 10)
7/5

property denom

property denominator

classmethod new(p, q)

property numer

property numerator

p

parent()

q

modelparameters.sympy.polys.domains.pythonrationalfield module

Implementation of PythonRationalField class.

class modelparameters.sympy.polys.domains.pythonrationalfield.PythonRationalField

Bases: RationalField

Rational field based on Python rational number type.

alias = 'QQ_python'

denom(a)

Returns denominator of a.

dtype

alias of PythonRational

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s Rational to dtype.

get_ring()

Returns ring associated with self.

numer(a)

Returns numerator of a.

one = 1

to_sympy(a)

Convert a to a SymPy object.

zero = 0

modelparameters.sympy.polys.domains.quotientring module

Implementation of QuotientRing class.

class modelparameters.sympy.polys.domains.quotientring.QuotientRingElement(ring, data)

Bases: object

Class representing elements of (commutative) quotient rings.

Attributes:

• ring - containing ring

• data - element of ring.ring (i.e. base ring) representing self

modelparameters.sympy.polys.domains.rationalfield module

Implementation of RationalField class.

class modelparameters.sympy.polys.domains.rationalfield.RationalField

Bases: Field, CharacteristicZero, SimpleDomain

General class for rational fields.

algebraic_field(*extension)

Returns an algebraic field, i.e. mathbb{Q}(alpha, ldots).

from_AlgebraicField(a, K0)

Convert a ANP object to dtype.

has_assoc_Field = True

has_assoc_Ring = True

is_Numerical = True

is_QQ = True

is_RationalField = True

rep = 'QQ'

modelparameters.sympy.polys.domains.realfield module

Implementation of RealField class.

class modelparameters.sympy.polys.domains.realfield.RealField(prec=53, dps=None, tol=None)

Bases: Field, CharacteristicZero, SimpleDomain

Real numbers up to the given precision.

almosteq(a, b, tolerance=None)

Check if a and b are almost equal.

property dps

from_ComplexField(element, base)

Convert a complex element to dtype.

from_QQ_gmpy(element, base)

Convert a GMPY mpq object to dtype.

from_QQ_python(element, base)

Convert a Python Fraction object to dtype.

from_RealField(element, base)

Convert a real element object to dtype.

from_ZZ_gmpy(element, base)

Convert a GMPY mpz object to dtype.

from_ZZ_python(element, base)

Convert a Python int object to dtype.

from_sympy(expr)

Convert SymPy’s number to dtype.

gcd(a, b)

Returns GCD of a and b.

get_exact()

Returns an exact domain associated with self.

get_ring()

Returns a ring associated with self.

has_assoc_Field = True

has_assoc_Ring = False

property has_default_precision

is_Exact = False

is_Numerical = True

is_PID = False

is_RR = True

is_RealField = True

lcm(a, b)

Returns LCM of a and b.

property precision

rep = 'RR'

to_rational(element, limit=True)

Convert a real number to rational number.

to_sympy(element)

Convert element to SymPy number.

property tolerance

modelparameters.sympy.polys.domains.ring module

Implementation of Ring class.

class modelparameters.sympy.polys.domains.ring.Ring

Bases: Domain

Represents a ring domain.

denom(a)

Returns denominator of a.

div(a, b)

Division of a and b, implies __divmod__.

exquo(a, b)

Exact quotient of a and b, implies __floordiv__.

free_module(rank)

Generate a free module of rank rank over self.

>>> from ...abc import x
>>> from ... import QQ
>>> QQ.old_poly_ring(x).free_module(2)
QQ[x]**2

get_ring()

Returns a ring associated with self.

ideal(*gens)

Generate an ideal of self.

>>> from ...abc import x
>>> from ... import QQ
>>> QQ.old_poly_ring(x).ideal(x**2)
<x**2>

invert(a, b)

Returns inversion of a mod b.

is_Ring = True

is_unit(a)

numer(a)

Returns numerator of a.

quo(a, b)

Quotient of a and b, implies __floordiv__.

quotient_ring(e)

Form a quotient ring of self.

Here e can be an ideal or an iterable.

>>> from ...abc import x
>>> from ... import QQ
>>> QQ.old_poly_ring(x).quotient_ring(QQ.old_poly_ring(x).ideal(x**2))
QQ[x]/<x**2>
>>> QQ.old_poly_ring(x).quotient_ring([x**2])
QQ[x]/<x**2>

The division operator has been overloaded for this:

>>> QQ.old_poly_ring(x)/[x**2]
QQ[x]/<x**2>

rem(a, b)

Remainder of a and b, implies __mod__.

revert(a)

Returns a**(-1) if possible.

modelparameters.sympy.polys.domains.simpledomain module

Implementation of SimpleDomain class.

class modelparameters.sympy.polys.domains.simpledomain.SimpleDomain

Bases: Domain

Base class for simple domains, e.g. ZZ, QQ.

inject(*gens)

Inject generators into this domain.

is_Simple = True

Module contents

Implementation of mathematical domains.

class modelparameters.sympy.polys.domains.AlgebraicField(dom, *ext)

Bases: Field, CharacteristicZero, SimpleDomain

A class for representing algebraic number fields.

algebraic_field(*extension)

Returns an algebraic field, i.e. mathbb{Q}(alpha, ldots).

denom(a)

Returns denominator of a.

dtype

alias of ANP

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s expression to dtype.

get_ring()

Returns a ring associated with self.

has_assoc_Field = True

has_assoc_Ring = False

is_Algebraic = True

is_AlgebraicField = True

is_Numerical = True

is_negative(a)

Returns True if a is negative.

is_nonnegative(a)

Returns True if a is non-negative.

is_nonpositive(a)

Returns True if a is non-positive.

is_positive(a)

Returns True if a is positive.

new(element)

numer(a)

Returns numerator of a.

to_sympy(a)

Convert a to a SymPy object.

class modelparameters.sympy.polys.domains.ComplexField(prec=53, dps=None, tol=None)

Bases: Field, CharacteristicZero, SimpleDomain

Complex numbers up to the given precision.

almosteq(a, b, tolerance=None)

Check if a and b are almost equal.

property dps

from_ComplexField(element, base)

Convert a complex element to dtype.

from_QQ_gmpy(element, base)

Convert a GMPY mpq object to dtype.

from_QQ_python(element, base)

Convert a Python Fraction object to dtype.

from_RealField(element, base)

Convert a real element object to dtype.

from_ZZ_gmpy(element, base)

Convert a GMPY mpz object to dtype.

from_ZZ_python(element, base)

Convert a Python int object to dtype.

from_sympy(expr)

Convert SymPy’s number to dtype.

gcd(a, b)

Returns GCD of a and b.

get_exact()

Returns an exact domain associated with self.

get_ring()

Returns a ring associated with self.

has_assoc_Field = True

has_assoc_Ring = False

property has_default_precision

is_CC = True

is_ComplexField = True

is_Exact = False

is_Numerical = True

lcm(a, b)

Returns LCM of a and b.

property precision

rep = 'CC'

to_sympy(element)

Convert element to SymPy number.

property tolerance

class modelparameters.sympy.polys.domains.Domain

Bases: object

Represents an abstract domain.

abs(a)

Absolute value of a, implies __abs__.

add(a, b)

Sum of a and b, implies __add__.

algebraic_field(*extension)

Returns an algebraic field, i.e. K(alpha, ldots).

alias = None

almosteq(a, b, tolerance=None)

Check if a and b are almost equal.

characteristic()

Return the characteristic of this domain.

cofactors(a, b)

Returns GCD and cofactors of a and b.

convert(element, base=None)

Convert element to self.dtype.

convert_from(element, base)

Convert element to self.dtype given the base domain.

denom(a)

Returns denominator of a.

div(a, b)

Division of a and b, implies something.

dtype = None

evalf(a, prec=None, **options)

Returns numerical approximation of a.

exquo(a, b)

Exact quotient of a and b, implies something.

frac_field(*symbols, **kwargs)

Returns a fraction field, i.e. K(X).

from_AlgebraicField(a, K0)

Convert an algebraic number to dtype.

from_ComplexField(a, K0)

Convert a complex element to dtype.

from_ExpressionDomain(a, K0)

Convert a EX object to dtype.

from_FF_gmpy(a, K0)

Convert ModularInteger(mpz) to dtype.

from_FF_python(a, K0)

Convert ModularInteger(int) to dtype.

from_FractionField(a, K0)

Convert a rational function to dtype.

from_GeneralizedPolynomialRing(a, K0)

from_GlobalPolynomialRing(a, K0)

Convert a polynomial to dtype.

from_PolynomialRing(a, K0)

Convert a polynomial to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a real element object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert a SymPy object to dtype.

gcd(a, b)

Returns GCD of a and b.

gcdex(a, b)

Extended GCD of a and b.

get_exact()

Returns an exact domain associated with self.

get_field()

Returns a field associated with self.

get_ring()

Returns a ring associated with self.

half_gcdex(a, b)

Half extended GCD of a and b.

has_CharacteristicZero = False

property has_Field

property has_Ring

has_assoc_Field = False

has_assoc_Ring = False

imag(a)

inject(*symbols)

Inject generators into this domain.

invert(a, b)

Returns inversion of a mod b, implies something.

is_Algebraic = False

is_AlgebraicField = False

is_CC = False

is_ComplexField = False

is_Composite = False

is_EX = False

is_Exact = True

is_FF = False

is_Field = False

is_FiniteField = False

is_Frac = False

is_FractionField = False

is_IntegerRing = False

is_Numerical = False

is_PID = False

is_Poly = False

is_PolynomialRing = False

is_QQ = False

is_RR = False

is_RationalField = False

is_RealField = False

is_Ring = False

is_Simple = False

is_SymbolicDomain = False

is_ZZ = False

is_negative(a)

Returns True if a is negative.

is_nonnegative(a)

Returns True if a is non-negative.

is_nonpositive(a)

Returns True if a is non-positive.

is_one(a)

Returns True if a is one.

is_positive(a)

Returns True if a is positive.

is_zero(a)

Returns True if a is zero.

lcm(a, b)

Returns LCM of a and b.

log(a, b)

Returns b-base logarithm of a.

map(seq)

Rersively apply self to all elements of seq.

mul(a, b)

Product of a and b, implies __mul__.

n(a, prec=None, **options)

Returns numerical approximation of a.

neg(a)

Returns a negated, implies __neg__.

new(*args)

normal(*args)

numer(a)

Returns numerator of a.

of_type(element)

Check if a is of type dtype.

old_frac_field(*symbols, **kwargs)

Returns a fraction field, i.e. K(X).

old_poly_ring(*symbols, **kwargs)

Returns a polynomial ring, i.e. K[X].

one = None

poly_ring(*symbols, **kwargs)

Returns a polynomial ring, i.e. K[X].

pos(a)

Returns a positive, implies __pos__.

pow(a, b)

Raise a to power b, implies __pow__.

quo(a, b)

Quotient of a and b, implies something.

real(a)

rem(a, b)

Remainder of a and b, implies __mod__.

rep = None

revert(a)

Returns a**(-1) if possible.

sqrt(a)

Returns square root of a.

sub(a, b)

Difference of a and b, implies __sub__.

to_sympy(a)

Convert a to a SymPy object.

property tp

unify(K1, symbols=None)

Construct a minimal domain that contains elements of K0 and K1.

Known domains (from smallest to largest):

• GF(p)

• ZZ

• QQ

• RR(prec, tol)

• CC(prec, tol)

• ALG(a, b, c)

• K[x, y, z]

• K(x, y, z)

• EX

unify_with_symbols(K1, symbols)

zero = None

class modelparameters.sympy.polys.domains.ExpressionDomain

Bases: Field, CharacteristicZero, SimpleDomain

A class for arbitrary expressions.

class Expression(ex)

Bases: PicklableWithSlots

An arbitrary expression.

as_expr()

denom()

ex

gcd(g)

lcm(g)

numer()

simplify(ex)

denom(a)

Returns denominator of a.

dtype

alias of Expression

from_ExpressionDomain(a, K0)

Convert a EX object to dtype.

from_FractionField(a, K0)

Convert a DMF object to dtype.

from_PolynomialRing(a, K0)

Convert a DMP object to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s expression to dtype.

gcd(a, b)

Returns GCD of a and b.

This definition of GCD over fields allows to clear denominators in primitive().

>>> from ..domains import QQ
>>> from ... import S, gcd, primitive
>>> from ...abc import x

>>> QQ.gcd(QQ(2, 3), QQ(4, 9))
2/9
>>> gcd(S(2)/3, S(4)/9)
2/9
>>> primitive(2*x/3 + S(4)/9)
(2/9, 3*x + 2)

get_field()

Returns a field associated with self.

get_ring()

Returns a ring associated with self.

has_assoc_Field = True

has_assoc_Ring = False

is_EX = True

is_SymbolicDomain = True

is_negative(a)

Returns True if a is negative.

is_nonnegative(a)

Returns True if a is non-negative.

is_nonpositive(a)

Returns True if a is non-positive.

is_positive(a)

Returns True if a is positive.

lcm(a, b)

Returns LCM of a and b.

>>> from ..domains import QQ
>>> from ... import S, lcm

>>> QQ.lcm(QQ(2, 3), QQ(4, 9))
4/3
>>> lcm(S(2)/3, S(4)/9)
4/3

numer(a)

Returns numerator of a.

one = EX(1)

rep = 'EX'

to_sympy(a)

Convert a to a SymPy object.

zero = EX(0)

modelparameters.sympy.polys.domains.FF

alias of PythonFiniteField

modelparameters.sympy.polys.domains.FF_gmpy

alias of GMPYFiniteField

modelparameters.sympy.polys.domains.FF_python

alias of PythonFiniteField

class modelparameters.sympy.polys.domains.FiniteField(mod, dom=None, symmetric=True)

Bases: Field, SimpleDomain

General class for finite fields.

characteristic()

Return the characteristic of this domain.

dom = None

from_FF_gmpy(a, K0=None)

Convert ModularInteger(mpz) to dtype.

from_FF_python(a, K0=None)

Convert ModularInteger(int) to dtype.

from_QQ_gmpy(a, K0=None)

Convert GMPY’s mpq to dtype.

from_QQ_python(a, K0=None)

Convert Python’s Fraction to dtype.

from_RealField(a, K0)

Convert mpmath’s mpf to dtype.

from_ZZ_gmpy(a, K0=None)

Convert GMPY’s mpz to dtype.

from_ZZ_python(a, K0=None)

Convert Python’s int to dtype.

from_sympy(a)

Convert SymPy’s Integer to SymPy’s Integer.

get_field()

Returns a field associated with self.

has_assoc_Field = True

has_assoc_Ring = False

is_FF = True

is_FiniteField = True

is_Numerical = True

mod = None

rep = 'FF'

to_sympy(a)

Convert a to a SymPy object.

class modelparameters.sympy.polys.domains.FractionField(domain_or_field, symbols=None, order=None)

Bases: Field, CompositeDomain

A class for representing multivariate rational function fields.

denom(a)

Returns denominator of a.

factorial(a)

Returns factorial of a.

from_AlgebraicField(a, K0)

Convert an algebraic number to dtype.

from_FractionField(a, K0)

Convert a rational function to dtype.

from_PolynomialRing(a, K0)

Convert a polynomial to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s expression to dtype.

get_ring()

Returns a field associated with self.

has_assoc_Field = True

has_assoc_Ring = True

is_Frac = True

is_FractionField = True

is_negative(a)

Returns True if LC(a) is negative.

is_nonnegative(a)

Returns True if LC(a) is non-negative.

is_nonpositive(a)

Returns True if LC(a) is non-positive.

is_positive(a)

Returns True if LC(a) is positive.

new(element)

numer(a)

Returns numerator of a.

property one

property order

to_sympy(a)

Convert a to a SymPy object.

property zero

modelparameters.sympy.polys.domains.GF

alias of PythonFiniteField

class modelparameters.sympy.polys.domains.GMPYFiniteField(mod, symmetric=True)

Bases: FiniteField

Finite field based on GMPY integers.

alias = 'FF_gmpy'

class modelparameters.sympy.polys.domains.GMPYIntegerRing

Bases: IntegerRing

Integer ring based on GMPY’s mpz type.

alias = 'ZZ_gmpy'

dtype

alias of GMPYInteger

factorial(a)

Compute factorial of a.

from_FF_gmpy(a, K0)

Convert ModularInteger(mpz) to GMPY’s mpz.

from_FF_python(a, K0)

Convert ModularInteger(int) to GMPY’s mpz.

from_QQ_gmpy(a, K0)

Convert GMPY mpq to GMPY’s mpz.

from_QQ_python(a, K0)

Convert Python’s Fraction to GMPY’s mpz.

from_RealField(a, K0)

Convert mpmath’s mpf to GMPY’s mpz.

from_ZZ_gmpy(a, K0)

Convert GMPY’s mpz to GMPY’s mpz.

from_ZZ_python(a, K0)

Convert Python’s int to GMPY’s mpz.

from_sympy(a)

Convert SymPy’s Integer to dtype.

gcd(a, b)

Compute GCD of a and b.

gcdex(a, b)

Compute extended GCD of a and b.

lcm(a, b)

Compute LCM of a and b.

one = <modelparameters.sympy.polys.domains.groundtypes.GMPYInteger object>

sqrt(a)

Compute square root of a.

to_sympy(a)

Convert a to a SymPy object.

tp

alias of GMPYInteger

zero = <modelparameters.sympy.polys.domains.groundtypes.GMPYInteger object>

class modelparameters.sympy.polys.domains.GMPYRationalField

Bases: RationalField

Rational field based on GMPY mpq class.

alias = 'QQ_gmpy'

denom(a)

Returns denominator of a.

div(a, b)

Division of a and b, implies __div__.

dtype

alias of GMPYRational

exquo(a, b)

Exact quotient of a and b, implies __div__.

factorial(a)

Returns factorial of a.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s Integer to dtype.

get_ring()

Returns ring associated with self.

numer(a)

Returns numerator of a.

one = <modelparameters.sympy.polys.domains.groundtypes.GMPYRational object>

quo(a, b)

Quotient of a and b, implies __div__.

rem(a, b)

Remainder of a and b, implies nothing.

to_sympy(a)

Convert a to a SymPy object.

tp

alias of GMPYRational

zero = <modelparameters.sympy.polys.domains.groundtypes.GMPYRational object>

class modelparameters.sympy.polys.domains.IntegerRing

Bases: Ring, CharacteristicZero, SimpleDomain

General class for integer rings.

algebraic_field(*extension)

Returns an algebraic field, i.e. mathbb{Q}(alpha, ldots).

from_AlgebraicField(a, K0)

Convert a ANP object to dtype.

get_field()

Returns a field associated with self.

has_assoc_Field = True

has_assoc_Ring = True

is_IntegerRing = True

is_Numerical = True

is_PID = True

is_ZZ = True

log(a, b)

Returns b-base logarithm of a.

rep = 'ZZ'

class modelparameters.sympy.polys.domains.PolynomialRing(domain_or_ring, symbols=None, order=None)

Bases: Ring, CompositeDomain

A class for representing multivariate polynomial rings.

factorial(a)

Returns factorial of a.

from_AlgebraicField(a, K0)

Convert an algebraic number to dtype.

from_FractionField(a, K0)

Convert a rational function to dtype.

from_PolynomialRing(a, K0)

Convert a polynomial to dtype.

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s expression to dtype.

gcd(a, b)

Returns GCD of a and b.

gcdex(a, b)

Extended GCD of a and b.

get_field()

Returns a field associated with self.

has_assoc_Field = True

has_assoc_Ring = True

is_Poly = True

is_PolynomialRing = True

is_negative(a)

Returns True if LC(a) is negative.

is_nonnegative(a)

Returns True if LC(a) is non-negative.

is_nonpositive(a)

Returns True if LC(a) is non-positive.

is_positive(a)

Returns True if LC(a) is positive.

lcm(a, b)

Returns LCM of a and b.

new(element)

property one

property order

to_sympy(a)

Convert a to a SymPy object.

property zero

class modelparameters.sympy.polys.domains.PythonFiniteField(mod, symmetric=True)

Bases: FiniteField

Finite field based on Python’s integers.

alias = 'FF_python'

class modelparameters.sympy.polys.domains.PythonIntegerRing

Bases: IntegerRing

Integer ring based on Python’s int type.

alias = 'ZZ_python'

dtype

alias of int

factorial(a)

Compute factorial of a.

from_FF_gmpy(a, K0)

Convert ModularInteger(mpz) to Python’s int.

from_FF_python(a, K0)

Convert ModularInteger(int) to Python’s int.

from_QQ_gmpy(a, K0)

Convert GMPY’s mpq to Python’s int.

from_QQ_python(a, K0)

Convert Python’s Fraction to Python’s int.

from_RealField(a, K0)

Convert mpmath’s mpf to Python’s int.

from_ZZ_gmpy(a, K0)

Convert GMPY’s mpz to Python’s int.

from_ZZ_python(a, K0)

Convert Python’s int to Python’s int.

from_sympy(a)

Convert SymPy’s Integer to dtype.

gcd(a, b)

Compute GCD of a and b.

gcdex(a, b)

Compute extended GCD of a and b.

lcm(a, b)

Compute LCM of a and b.

one = 1

sqrt(a)

Compute square root of a.

to_sympy(a)

Convert a to a SymPy object.

zero = 0

class modelparameters.sympy.polys.domains.PythonRationalField

Bases: RationalField

Rational field based on Python rational number type.

alias = 'QQ_python'

denom(a)

Returns denominator of a.

dtype

alias of PythonRational

from_QQ_gmpy(a, K0)

Convert a GMPY mpq object to dtype.

from_QQ_python(a, K0)

Convert a Python Fraction object to dtype.

from_RealField(a, K0)

Convert a mpmath mpf object to dtype.

from_ZZ_gmpy(a, K0)

Convert a GMPY mpz object to dtype.

from_ZZ_python(a, K0)

Convert a Python int object to dtype.

from_sympy(a)

Convert SymPy’s Rational to dtype.

get_ring()

Returns ring associated with self.

numer(a)

Returns numerator of a.

one = 1

to_sympy(a)

Convert a to a SymPy object.

zero = 0

modelparameters.sympy.polys.domains.QQ_gmpy

alias of GMPYRationalField

modelparameters.sympy.polys.domains.QQ_python

alias of PythonRationalField

class modelparameters.sympy.polys.domains.RationalField

Bases: Field, CharacteristicZero, SimpleDomain

General class for rational fields.

algebraic_field(*extension)

Returns an algebraic field, i.e. mathbb{Q}(alpha, ldots).

from_AlgebraicField(a, K0)

Convert a ANP object to dtype.

has_assoc_Field = True

has_assoc_Ring = True

is_Numerical = True

is_QQ = True

is_RationalField = True

rep = 'QQ'

class modelparameters.sympy.polys.domains.RealField(prec=53, dps=None, tol=None)

Bases: Field, CharacteristicZero, SimpleDomain

Real numbers up to the given precision.

almosteq(a, b, tolerance=None)

Check if a and b are almost equal.

property dps

from_ComplexField(element, base)

Convert a complex element to dtype.

from_QQ_gmpy(element, base)

Convert a GMPY mpq object to dtype.

from_QQ_python(element, base)

Convert a Python Fraction object to dtype.

from_RealField(element, base)

Convert a real element object to dtype.

from_ZZ_gmpy(element, base)

Convert a GMPY mpz object to dtype.

from_ZZ_python(element, base)

Convert a Python int object to dtype.

from_sympy(expr)

Convert SymPy’s number to dtype.

gcd(a, b)

Returns GCD of a and b.

get_exact()

Returns an exact domain associated with self.

get_ring()

Returns a ring associated with self.

has_assoc_Field = True

has_assoc_Ring = False

property has_default_precision

is_Exact = False

is_Numerical = True

is_PID = False

is_RR = True

is_RealField = True

lcm(a, b)

Returns LCM of a and b.

property precision

rep = 'RR'

to_rational(element, limit=True)

Convert a real number to rational number.

to_sympy(element)

Convert element to SymPy number.

property tolerance

modelparameters.sympy.polys.domains.ZZ_gmpy

alias of GMPYIntegerRing

modelparameters.sympy.polys.domains.ZZ_python

alias of PythonIntegerRing